Optimal. Leaf size=30 \[ -\frac {1}{8} x \sqrt {5-4 x^2}+\frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 222}
\begin {gather*} \frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right )-\frac {1}{8} x \sqrt {5-4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {5-4 x^2}} \, dx &=-\frac {1}{8} x \sqrt {5-4 x^2}+\frac {5}{8} \int \frac {1}{\sqrt {5-4 x^2}} \, dx\\ &=-\frac {1}{8} x \sqrt {5-4 x^2}+\frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 41, normalized size = 1.37 \begin {gather*} -\frac {1}{8} x \sqrt {5-4 x^2}+\frac {5}{16} i \log \left (-2 i x+\sqrt {5-4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.97, size = 22, normalized size = 0.73 \begin {gather*} -\frac {x \sqrt {5-4 x^2}}{8}+\frac {5 \text {ArcSin}\left [\frac {2 \sqrt {5} x}{5}\right ]}{16} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 23, normalized size = 0.77
method | result | size |
default | \(\frac {5 \arcsin \left (\frac {2 x \sqrt {5}}{5}\right )}{16}-\frac {x \sqrt {-4 x^{2}+5}}{8}\) | \(23\) |
risch | \(\frac {x \left (4 x^{2}-5\right )}{8 \sqrt {-4 x^{2}+5}}+\frac {5 \arcsin \left (\frac {2 x \sqrt {5}}{5}\right )}{16}\) | \(30\) |
meijerg | \(\frac {5 i \left (\frac {2 i \sqrt {\pi }\, x \sqrt {5}\, \sqrt {-\frac {4 x^{2}}{5}+1}}{5}-i \sqrt {\pi }\, \arcsin \left (\frac {2 x \sqrt {5}}{5}\right )\right )}{16 \sqrt {\pi }}\) | \(40\) |
trager | \(-\frac {x \sqrt {-4 x^{2}+5}}{8}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-4 x^{2}+5}+2 x \right )}{16}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 22, normalized size = 0.73 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 5} x + \frac {5}{16} \, \arcsin \left (\frac {2}{5} \, \sqrt {5} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 30, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 5} x - \frac {5}{16} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 5}}{2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 27, normalized size = 0.90 \begin {gather*} - \frac {x \sqrt {5 - 4 x^{2}}}{8} + \frac {5 \operatorname {asin}{\left (\frac {2 \sqrt {5} x}{5} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 33, normalized size = 1.10 \begin {gather*} -\frac {2}{16} x \sqrt {-4 x^{2}+5}+\frac {5}{16} \arcsin \left (\frac {2 x}{\sqrt {5}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 22, normalized size = 0.73 \begin {gather*} \frac {5\,\mathrm {asin}\left (\frac {2\,\sqrt {5}\,x}{5}\right )}{16}-\frac {x\,\sqrt {\frac {5}{4}-x^2}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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